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Quantum Channel Capacity of Very Noisy Channels

David P. DiVincenzo·Peter W. Shor·John A. Smolin
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Abstract

We present a family of additive quantum error-correcting codes whose capacities exceeds that of quantum random coding (hashing) for very noisy channels. These codes provide non-zero capacity in a depolarizing channel for fidelity parameters $f$ when $f> .80944$. Random coding has non-zero capacity only for $f>.81071$; by analogy to the classical Shannon coding limit, this value had previously been conjectured to be a lower bound. We use the method introduced by Shor and Smolin of concatenating a non-random (cat) code within a random code to obtain good codes. The cat code with block size five is shown to be optimal for single concatenation. The best known multiple-concatenated code we found has a block size of 25. We derive a general relation between the capacity attainable by these concatenation schemes and the coherent information of the inner code states.

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David P. DiVincenzo, Peter W. Shor, John A. Smolin. 1998-11-03. Quantum Channel Capacity of Very Noisy Channels. https://doi.org/10.1103/physreva.57.830

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